TITLE: "Surprising" examples of Markov chains
QUESTION [59 upvotes]: I am looking for examples of Markov Chains which are surprising in the following sense: a stochastic process $X_1,X_2,...$ which is "natural" but for which the Markov property is not obvious at first glance. For example, it could be that the natural definition of the process is in terms of some process $Y_1,Y_2,...$ which is Markovian and where $X_i=f(Y_i)$ for some function $f$. 
Let me give you an example which is slightly surprising, but not surprising enough for my taste. Suppose we have $n$ bins that are initially empty, and at each time step $t$ we throw a ball into one of the bins selected uniformly at random (and independently of previous time steps). Let $X_t$ be the number of empty bins at time $t$. Then $X_1,X_2,...$ form a Markov chain. 
Are there good, more surprising examples? I apologize if this question is vague.

REPLY [3 votes]: The Dyson Brownian motion. Take a random matrix whose entries perform independent standard Brownian motions, subject to the condition that the matrix stays Hermitian, so that entries above diagonal perform Brownian motions in the complex plane, the diagonal entries perform a BM on the real line, and the entries below the diagonal are complex conjugates of the ones above that diagonal. 
Then, the eigenvalue process $(\lambda_1(t),\dots,\lambda_n(t))$ of this random matrix is Markov. This is closely related to the example mentioned in Leonid Petrov's answer, but I think the reasons for this "surprising Markov property" are rather different in the two cases.